Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminflimsupd | Structured version Visualization version GIF version |
Description: If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminflimsupd.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climliminflimsupd.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climliminflimsupd.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
climliminflimsupd.4 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Ref | Expression |
---|---|
climliminflimsupd | ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminflimsupd.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
2 | 1 | feqmptd 6728 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
3 | 2 | fveq2d 6669 | . . . . 5 ⊢ (𝜑 → (lim inf‘𝐹) = (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))) |
4 | climliminflimsupd.2 | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | fvexi 6679 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 6980 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | liminfcl 42036 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V → (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ*) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ* |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ*) |
10 | 3, 9 | eqeltrd 2913 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) |
11 | nfv 1911 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
12 | climliminflimsupd.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 1 | ffvelrnda 6846 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | 13 | renegcld 11061 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
15 | 11, 12, 4, 14 | limsupvaluz4 42073 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)))) |
16 | climrel 14843 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel ⇝ ) |
18 | nfcv 2977 | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝐹 | |
19 | climliminflimsupd.4 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
20 | 12, 4, 1 | climlimsup 42033 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
21 | 19, 20 | mpbid 234 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
22 | 13 | recnd 10663 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
23 | 11, 18, 4, 12, 21, 22 | climneg 41883 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) |
24 | releldm 5809 | . . . . . . . . 9 ⊢ ((Rel ⇝ ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) | |
25 | 17, 23, 24 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) |
26 | 14 | fmpttd 6874 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)):𝑍⟶ℝ) |
27 | 12, 4, 26 | climlimsup 42033 | . . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ↔ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))))) |
28 | 25, 27 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) |
29 | climuni 14903 | . . . . . . 7 ⊢ (((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) | |
30 | 28, 23, 29 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) |
31 | 22 | negnegd 10982 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → --(𝐹‘𝑘) = (𝐹‘𝑘)) |
32 | 31 | mpteq2dva 5154 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
33 | 32, 2 | eqtr4d 2859 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = 𝐹) |
34 | 33 | fveq2d 6669 | . . . . . . 7 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = (lim inf‘𝐹)) |
35 | 34 | xnegeqd 41703 | . . . . . 6 ⊢ (𝜑 → -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = -𝑒(lim inf‘𝐹)) |
36 | 15, 30, 35 | 3eqtr3d 2864 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) = -𝑒(lim inf‘𝐹)) |
37 | 4, 12, 21, 13 | climrecl 14934 | . . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
38 | 37 | renegcld 11061 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) ∈ ℝ) |
39 | 36, 38 | eqeltrrd 2914 | . . . 4 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) ∈ ℝ) |
40 | xnegrecl2 41728 | . . . 4 ⊢ (((lim inf‘𝐹) ∈ ℝ* ∧ -𝑒(lim inf‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ) | |
41 | 10, 39, 40 | syl2anc 586 | . . 3 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) |
42 | 41 | recnd 10663 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℂ) |
43 | 37 | recnd 10663 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℂ) |
44 | 41 | rexnegd 41404 | . . 3 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) = -(lim inf‘𝐹)) |
45 | 36, 44 | eqtr2d 2857 | . 2 ⊢ (𝜑 → -(lim inf‘𝐹) = -(lim sup‘𝐹)) |
46 | 42, 43, 45 | neg11d 11003 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3495 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5550 Rel wrel 5555 ⟶wf 6346 ‘cfv 6350 ℝcr 10530 ℝ*cxr 10668 -cneg 10865 ℤcz 11975 ℤ≥cuz 12237 -𝑒cxne 12498 lim supclsp 14821 ⇝ cli 14835 lim infclsi 42024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-ico 12738 df-fl 13156 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-liminf 42025 |
This theorem is referenced by: climliminf 42079 climliminflimsup 42081 climliminflimsup2 42082 xlimliminflimsup 42135 |
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